Optical and Photoelectric Analog of the Eye

Otto H. Schade

doi:10.1364/JOSA.46.000721

Journal of the Optical Society of America
Vol. 46,
Issue 9,
pp. 721-739
(1956)
•https://doi.org/10.1364/JOSA.46.000721

Optical and Photoelectric Analog of the Eye

Otto H. Schade

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Otto H. Schade, "Optical and Photoelectric Analog of the Eye," J. Opt. Soc. Am. 46, 721-739 (1956)

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Abstract

A photoelectric analog of the visual system is constructed in conformance with anatomical data. The analog has the form of a color television camera chain feeding electrical signals to a “computer” (the brain). Evaluation of characteristics is limited to elements preceding the computer, and particularly to the “luminance channel” of the color system.

The primary photoelectric transfer characteristics ${\bar{n}}_{1}$ =f(E_r) of the receptors (rods and cones) are computed as a function of retinal illumination (E_r) from threshold signal-to-noise ratios in the effective image area of point sources, disks, and other test objects. The effective image area, which is the convolution of the object area with the sampling area of the visual system, is determined from its Fourier spectrum. The constants of the transfer functions are established from the optical constants of the eye, its storage-time function, and the maximum transfer ratio of statistical units of the rod and cone systems. There is little room for variation of constants, if they are to remain in agreement with observed values.

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Equations (50)

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B (foot-lambert)	E_r (trolands)
10⁻⁶	1.35×10⁻⁴
4×10⁻⁶	5.4×10⁻⁴
10⁻⁵	1.32×10⁻³
4×10⁻⁵	5.1×10⁻³
10⁻⁴	1.21×10⁻²
4×10⁻⁴	4.5×10⁻²
10⁻³	1.08×10⁻¹
4×10⁻³	3.8×10⁻¹
10⁻²	8.7×10⁻¹
4×10⁻²	3.05
10⁻¹	7.0
4×10⁻¹	23.5
1	51.
4	167.
10	350.
40	1000.
100	2050.
400	5900.
1000	12 000.

Rods			Cones						Rods	Cones
logE_r	N_e₍_e₊₎	N_e₍_r₎	N_e₍_e₊₎	N_e₍_r₎	N_e₍_e₋₎	(W−/W)	f/	N_e₍_p.L_.)	N_e₍_L₎	N_e₍_L₎
−4.0	6.0	7.4							10
−3.4	6.7	8.9					2.4	381	10
−3.0	7.2	10.5	8.9	19.1	0.33	0.43			10
−2.4	8.0	13.1	9.1	19.1	0.4	0.5			10.2
−2.0	8.9	15.9	9.35	19.5	0.48	0.55	2.5	366	10.8
−1.4	10.2	20.4	10.75	22	0.61	0.60			11.7
−1.0	11.4	23.5	11.8	24.6	0.74	0.645	2.66	344	13.0
−0.4	14.5	31	14.5	31	1.07	0.69			16.4
0.	16.6	35.6	16.6	37.2	1.5	0.71	2.95	310	18.7	18.7
0.6	21.4	46.	21.4	48.	1.7	0.74			24.2	24.2
1.0	24	52.5	25.7	59	1.78	0.76	3.22	284		29.
1.6	30	62.5	34.7	79.5	1.86	0.78				39.6
2.0	34	67.6	40.7	98	1.9	0.80	4.1	223		46.4
2.6	38	77.5	50.	123.5	2.0	0.815				57.8
3.0	40	79.5	56.5	141.5	2.1	0.83	5.5	166.5		66.0
3.6		79.5	66	166.	2.3	0.83				79.0
4.0			71	182.	2.46	0.85	8.	114.		86.0
4.6			71	191.						85.5
Column #	1	2	3	4	5	6	7	8	9	10
	$\begin{array}{l} 1, & 3 & Equivalent passband of eye \\ 2, & 4 & Equivalent passband of retinal elements \end{array}} (normal positive component)$
	5 Equivalent passband of feedback image (negative image component)
	6 Relative energy of negative to positive point-image (see Fig. 8)
	7 f/ number of eye
	8 Equivalent passband of perfect lens for white light (luminosity weighting) (see reference 6)
	9, 10 Equivalent passband of lens (eye)

Acuity					Rods	Cones	Rods	Cones
logE_r	N_e₍₀₎	N_e₍_e₎	N_e(3)	${\dot{γ}}_{1}^{2} {\bar{n}}_{1}$	E_r/ΔE_r		${\dot{γ}}_{1}^{2} {\bar{n}}_{1}$	γ₁² ${\bar{n}}_{1}$
−4.0	0.95	6.0	0.94	46.2	1.3		122.
−3.75	1.83	6.3	1.75	185.
−3.4	3.5	6.7	3.1	790.	3.5		1.11×10³
−3.0	5.9	7.2	4.57	2.92×10³	6.0	1.0	3.76×10³
−2.4	10.5	8.0	6.35	1.51×10⁴	11.	3.9	1.56×10⁴
−2.0	14.5	8.9	7.6	3.85×10⁴	15.	7.6	3.6×10⁴
−1.4	22.5	10.2	9.3	1.48×10⁵	23.6	18	1.18×10⁵
−1.0	30.	11.4	10.7	3.56×10⁵	31.	31	2.54×10⁵	2.54×10⁵
−0.4	51.	14.5	13.9	1.77×10⁶	44.	59	8.2×10⁵	1.48×10⁶
0.	79.5	16.6	16.3	7.55×10⁶	50.	95	1.39×10⁶	5×10⁶
0.6	141.5	21.4	21.2	4.5×410⁷	58.	159	3.1×10⁶	2.3×10⁷
1.0	200	25.7	25.2	1.28×210⁸	62.	210	4.5×10⁶	5.9×10⁷
1.6	300	34.7	34.6	3.41×10⁸	65.	282	7.7×10⁶	1.94×10⁸
2.0	356	40.7	40.5	4.9×10⁸	65.	316	0.9×10⁶	3.35×10⁸
2.6	428	50.	49.7	6.9×10⁸		355		6.4×10⁸
3.0	450	56.5	56.5	6.5×10⁸		363		8.5×10⁸
3.6	490	66.	65.5	6.8×10⁸		375		1.11×10⁹
4.0	500	71.	69.5	6.55×10⁸		372		1.41×10⁹
						370		1.4×10⁹
${\dot{γ}}_{1}^{2} {\bar{n}}_{1}$ computed with Eq. (20) K=2.25, T−0.1, C₀=1					${\dot{γ}}_{1}^{2} {\bar{n}}_{1}$ computed with Eq. (22) $K = 0.45 c^{\frac{1}{2}}$ , T=0.1, N_e₍_e₎ from Table II

Rods									Cones
log E_r	log T	log ${\dot{γ}}_{1}^{2} {\bar{n}}_{1}$	log ${\bar{n}}_{1}$	${\dot{γ}}_{1}$	log s	log ${\bar{s}}_{3}$		log ${\bar{n}}_{2}$	log T	log ${\dot{γ}}_{1}^{2}$ n₁	log ${\bar{n}}_{1}$	${\dot{γ}}_{1}$	log s	log ${\bar{s}}_{3}$	log ${\bar{n}}_{2}$
−4.0	−0.72	1.54	2.5	1.	0.0	0.20*	0.72	2.46
−3.4	−0.80	2.75	3.1	1.	0.56	0.795*	1.0	2.94
−3.0	−0.86	3.33	3.5	1.	0.87	1.11*	1.2	3.25	−0.89		3.9	1.0	0.0	−0.07	2.49
−2.4	−0.92	4.04	4.1	1.	1.22	1.44*	1.47	3.71	−0.92		4.5	1.0	0.59	0.53	3.09
−2.0	−0.96	4.48	4.5	1.	1.43	1.64		4.04	−0.96		4.9	1.0	0.93	0.93	3.51
−1.4	−0.98	5.05	5.09	0.95	1.70	1.86		4.48	−0.98		5.5	1.0	1.40	1.44	4.12
−1.0	−0.99	5.41	5.48	0.90	1.85	2.0		4.74	−0.99		5.9	1.0	1.66	1.72	4.50
−0.4	−1.00	5.87	6.0	0.85	2.05	2.17		5.15	−1.0	6.17	6.5	1.0	2.04	2.06	5.04
0.0	−1.00	6.12	6.3	0.80	2.17	2.29		5.39	−1.0	6.82	6.9	1.0	2.25	2.27	5.14
0.6	−1.01	6.41	6.75	0.68	2.33	2.42		5.75	−1.03	7.5	7.5	1.0	2.50	2.55	5.91
1.0	−1.02	6.59	7.01	0.60	2.41	2.51		5.95	−1.06	7.9	7.9	1.0	2.67	2.71	6.25
1.6	−1.03	6.80	7.37	0.52	2.49	2.61		6.20	−1.13	8.42	8.5	0.95	2.875	2.93	6.73
2.0	−1.04	6.91	7.59	0.46	2.55	2.68		6.35	−1.20	8.72	8.83	0.88	3.00	3.06	7.04
2.6	−1.04		7.85		2.60	2.77		6.55	−1.27	9.01	9.31	0.71	3.156	3.25	7.43
3.0	−1.04		7.95		2.65	2.81		6.61	−1.30	9.13	9.57	0.60	3.245	3.35	7.65
3.6						2.87		6.67	−1.32	9.26	9.91	0.47	3.335	3.46	7.90
4.0									−1.33	9.31	10.11	0.40	3.392	3.53	8.05
4.6									−1.33	9.34	10.3	0.33	3.45	3.59
	1	2	3	4	5	6		7	1	2	3	4	5	6	7
		Column
		1	Storage time (see Fig. 14)
		2	Computed with Eq. (20) or (22)
		3	Primary photoelectric process
		4	Gamma of 3
		5	Sensation units (s* including effect of dark current) per picture frame
		6	Signal units per second in optic nerve [Eq. (25)], values* contain dark current effect
		7	Receptor output signals [Eq. (27)]

V	0.1	0.2	0.5	1	2	3	4	5	6	7	8	9	10
Y	0.12	0.237	0.58	1.21	3.126	6.56	12.0	19.77	30.05	43.06	59.1	78.66	102.56

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Journal of the Optical Society of America